Fig. 2. Multi-level enzyme cascade. Abbreviations: S, signal; E
i
and E
i
, active and inactive forms, respectively, of the enzyme at level i; T and T, active and inactive forms, respectively, of
the target.
This matrix cannot be block-diagonalized, while the corresponding null-space matrix can. The latter
has formally the same form as given in Eq. 7. By the formalism developed in Schuster and
Schuster 1992, it can be shown that reactions 3 and 4 would not exert any control on the flux
through reactions 1 and 2 which is, in fact, the glycolytic
flux if
fructose-2,6-bisphosphate F2,6P
2
were no effector of enzymes 1 and 2. However, F2,6P
2
is known to be a potent activator of 6-phosphofructokinase in many cell types and to
play an important role in the regulation of glycol- ysis Yuan et al., 1990; Lefebvre et al., 1996. This
will be analyzed in more detail at the end of the following section.
In a mutant where activation by F2,6P
2
is absent, reactions 1 and 2 do exert a dictatorial control on
the F2,6P
2
cycle irrespective of the values of kinetic parameters. Note that the feature of flux control
insusceptibility is asymmetric; it may happen that a subsystem 1 is able to control a subsystem 2 while
the latter cannot control the former.
If not only information is available about which elasticities are zero and which are not, but also the
signs of the elasticities are known, qualitative conclusions about the sign pattern of control coeffi-
cients can be drawn Lundy and Sen, 1995. Gen- erally, however, only the signs of a subset of the
control coefficients can be determined in this situ- ation.
5. Control of signal transduction
Consider a multi-level enzyme cascade as shown in Fig. 2. An important quantity is here the
response coefficient of the cellular target, T, to the signal, S:
R
S T
= d lnT
d lnS .
10 The signal may be a growth factor, hormone,
cytokine or neurotransmitter. When a single cascade module is considered ‘in
isolation’, the response of a signaling protein in this module E
i
to the immediately preceding module E
i − 1
is quantified as the intrinsic response coeffi- cient:
Fig. 3. Scheme of the fructose-2,6-bisphosphate cycle. En- zymes: E
1
, phosphoglucoisomerase EC 5.3.1.9; E
2
, 6-phos- phofructokinase EC 2.7.1.11; E
3
, fructose-2,6-bisphosphate 2-phosphatase EC 3.1.3.46; E
4
, 6-phosphofructo-2-kinase EC
2.7.1.105. Metabolites:
F6P, fructose-6-phosphate;
F2,6P
2
, fructose-2,6-bisphosphate.
the concentrations of enzyme-enzyme complexes forming in the catalytic cycles similar to the
complex ES in Fig. 1B are so high that they cannot be neglected enzyme sequestration, a lower level
enzyme can control a higher level cycle Schuster and Schuster, 1992.
In hierarchical systems without feedback, dicta- torial control is intuitively clear. The situation is
more complex in certain systems that are stoichio- metrically connected. Consider the example of the
fructose-2,6-bisphosphate cycle Fig. 3, which is a frequently present futile cycle Yuan et al., 1990;
Lefebvre et al., 1996. The stoichiometry matrix to this scheme reads
N =
1 −
1 1
− 1
− 1
1 .
9
r
i − 1 i
= d lnE
i
d lnE
i − 1
Ej=const
. 11
For linear signal transduction pathways without feedback, it has been shown Kholodenko et al.,
1997 that the response coefficients of each cas- cade level multiply to give the total response of a
target to a signal:
R
S T
= r
S 1
· r
1 2
· … · r
n − 1 T
. 12
Here, it is assumed that the sequestration of molecules at each level by the enzymes at the
successive level can be neglected Kholodenko et al., 1992. In the cases of branched pathways in
the cascade and of pathways with feedback, ana- lytical formulas with a more complex structure
can be derived Kholodenko et al., 1997.
From Eq. 12, it can be seen that merely having more cascade levels greatly enhances the
sensitivity of the target. As the elasticity coeffi- cient can be considered as a sort of reaction order,
it can be seen that when each elasticity is larger than one, high overall reaction orders and, hence,
a strong cooperativity, can be obtained Brown et al., 1997.
The question arises as to how the local response depends on the kinetics of its own level. Let us
consider a simple kinasephosphatase cycle, where the phosphorylated form E
i
is the active form, and the dephosphorylated form E
i
is inactive. Other forms of covalent enzyme modification
such as acylation, adenylylation, or uridylylation can be described similarly. Kinase and phos-
phatase reactions can often be assumed nearly irreversible, so that the kinase depends only on its
substrate E
i
and is catalysed by E
i − 1
and the phosphatase depends only on the form E
i
. Thus, only the two elasticities o
E
i ’
kin
and o
E
i
phos
are needed. The following formula can be derived Small and
Fell, 1990; Kholodenko et al., 1997:
r
i − 1 i
= 1
o
E
i
phos
+ o
E
i ’
kin
E
i
E
i
. 13
When both protein kinase and phosphatase follow Michaelis – Menten kinetics and are far from
equilibrium and product insensitive, the elastic- ities read
o
E
i
kin
= 1
1 + E
i
K
mKin
, o
E
i
phos
= 1
1 + E
i
K
mPhos
. 14
Here K
mKin
and K
mPhos
are the respective Michaelis constants. If the kinase and phos-
phatase are nearly saturated with their substrates i.e. with the enzymes on the next lower level, the
elasticities are very low. This implies that the local response coefficient is very high. Therefore, even a
monocyclic cascade can constitute a highly effec- tive onoff switching device Szedlacsek et al.,
1992; Acerenza, 1996. This phenomenon is called ‘zero-order
ultrasensitivity’ Goldbeter
and Koshland, 1984.
We will now show that similar considerations apply to stoichiometrically connected systems the
null-space matrix of which can be block-diagonal- ized. In particular, we will again consider the
F2,6P
2
cycle Fig. 3. Let S
1
and S
2
denote the concentrations of F6P and F2,6P
2
. Furthermore, A is to denote the level of an effector of 6-phos-
phofructo-2-kinase E
4
. As most kinase and phosphatase reactions, steps 3 and 4 in Fig. 3 can
be assumed to be nearly irreversible. We shall also assume them to be product insensitive. Accord-
ingly, the elasticities o
31
and o
42
can be neglected. At steady state, we have
6
1
S
1
= 6
2
S
1
, S
2
, 6
3
S
2
= 6
4
S
1
, A. 15
As this equation must still hold if the value of A is changed, we can write, by the chain rule of
differentiation
6
1
S
1
S
1
A =
6
2
S
1
S
1
A +
6
2
S
2
S
2
A ,
6
3
S
2
S
2
A =
6
4
S
1
S
1
A +
6
4
A .
16 After normalization, these equations can be writ-
ten as o
11
R
A S
1
= o
21
R
A S
1
+ o
22
R
A S
2
, o
32
R
A S
2
= o
41
R
A S
1
+ p
4A
17 where R
A S
1
and R
A S
2
denote the response coeffi- cients of S
1
and S
2
, respectively, with respect to the effector A, and p
4A
stands for the parameter elasticity of reaction 4 with respect to A. Solving
the equation system 17 for the response coeffi- cients gives
R
A S
1
= o
22
p
4A
o
32
o
11
− o
21
− o
41
o
22
, R
A S
2
= p
4A
o
11
− o
21
o
32
o
11
− o
21
− o
41
o
22
. 18
These equations can also be derived from the block summation and block connectivity theo-
rems established by Kahn and Westerhoff 1991. It has been shown that these theorems hold even
if the system is stoichiometrically connected, pro- vided that the null-space matrix and link matrix
which expresses the conservation relations in the system can be block-diagonalized Heinrich and
Schuster, 1996.
It can be seen from Eq. 18 that an extremely high response of F2,6P
2
S
2
can be achieved if the two elasticities o
32
and o
41
are very low, that is, if enzymes 3 and 4 are nearly saturated with their
substrates. The effect of activating 6-phospho- fructo-2-kinase on the glycolytic flux, J
2
, can be quantified by a flux response coefficient,
R
A J
2
= o
21
R
A S
1
+ o
22
R
A S
21
= o
21
o
22
p
4A
o
32
o
11
− o
21
− o
41
o
22
. 19
Again, this effect is very high in the case of saturation of enzymes 3 and 4. Accordingly, the
phenomenon of zero-order ultra-sensitivity can occur even in systems that do not have a cascade
structure. An important difference to the cyclic systems studied earlier Goldbeter and Koshland,
1984; Small and Fell, 1990; Kholodenko et al., 1997 is that one substance involved in the cycle
F6P in our case is subject to turnover, so that the total amount of substance in the cycle is not
conserved.
6. Discussion
